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std::beta, std::betaf, std::betal

Defined in header `<cmath>`
	(1)
float beta ( float x, float y ); double beta ( double x, double y ); long double beta ( long double x, long double y );		(since C++17) (until C++23)
/* floating-point-type / beta( / floating-point-type / x, / floating-point-type */ y );		(since C++23)
float betaf( float x, float y );	(2)	(since C++17)
long double betal( long double x, long double y );	(3)	(since C++17)
Additional overloads
Defined in header `<cmath>`
template< class Arithmetic1, class Arithmetic2 > /* common-floating-point-type */ beta( Arithmetic1 x, Arithmetic2 y );	(A)	(since C++17)

1-3) Computes the Beta function of x and y. The library provides overloads of std::beta for all cv-unqualified floating-point types as the type of the parameters x and y. (since C++23)

A) Additional overloads are provided for all other combinations of arithmetic types.

Parameters

x, y

floating-point or integer values

Return value

If no errors occur, value of the beta function of x and y, that is \(\int_{0}^{1}{ {t}^{x-1}{(1-t)}^{y-1}\mathsf{d}t}\)∫1
0tx-1
(1-t)(y-1)
dt, or, equivalently, \(\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}\)Γ(x)Γ(y)/Γ(x+y) is returned.

Error handling

Errors may be reported as specified in math_errhandling.

If any argument is NaN, NaN is returned and domain error is not reported
The function is only required to be defined where both x and y are greater than zero, and is allowed to report a domain error otherwise.

Notes

Implementations that do not support C++17, but support ISO 29124:2010, provide this function if __STDCPP_MATH_SPEC_FUNCS__ is defined by the implementation to a value at least 201003L and if the user defines __STDCPP_WANT_MATH_SPEC_FUNCS__ before including any standard library headers.

Implementations that do not support ISO 29124:2010 but support TR 19768:2007 (TR1), provide this function in the header tr1/cmath and namespace std::tr1.

An implementation of this function is also available in boost.math.

std::beta(x, y) equals std::beta(y, x). When x and y are positive integers, std::beta(x, y) equals \(\frac{(x-1)!(y-1)!}{(x+y-1)!}\).

(x-1)!(y-1)!/(x+y-1)!. Binomial coefficients can be expressed in terms of the beta function: \(\binom{n}{k} = \frac{1}{(n+1)B(n-k+1,k+1)}\)⎛
⎜
⎝n
k⎞
⎟
⎠=1/(n+1)Β(n-k+1,k+1).

The additional overloads are not required to be provided exactly as (A). They only need to be sufficient to ensure that for their first argument num1 and second argument num2:

If num1 or num2 has type long double, then std::beta(num1, num2) has the same effect as std::beta(static_cast<long double>(num1), static_cast<long double>(num2)).
Otherwise, if num1 and/or num2 has type double or an integer type, then std::beta(num1, num2) has the same effect as std::beta(static_cast<double>(num1), static_cast<double>(num2)).
Otherwise, if num1 or num2 has type float, then std::beta(num1, num2) has the same effect as std::beta(static_cast<float>(num1), static_cast<float>(num2)).

(until C++23)

If num1 and num2 have arithmetic types, then std::beta(num1, num2) has the same effect as std::beta(static_cast</* common-floating-point-type */>(num1), static_cast</* common-floating-point-type */>(num2)), where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank between the types of num1 and num2, arguments of integer type are considered to have the same floating-point conversion rank as double.

If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the overloads provided.

(since C++23)

Example

#include <cassert>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <numbers>
#include <string>
 
long binom_via_beta(int n, int k)
{
    return std::lround(1 / ((n + 1) * std::beta(n - k + 1, k + 1)));
}
 
long binom_via_gamma(int n, int k)
{
    return std::lround(std::tgamma(n + 1) /
                      (std::tgamma(n - k + 1) * 
                       std::tgamma(k + 1)));
}
 
int main()
{
    std::cout << "Pascal's triangle:\n";
    for (int n = 1; n < 10; ++n)
    {
        std::cout << std::string(20 - n * 2, ' ');
        for (int k = 1; k < n; ++k)
        {
            std::cout << std::setw(3) << binom_via_beta(n, k) << ' ';
            assert(binom_via_beta(n, k) == binom_via_gamma(n, k));
        }
        std::cout << '\n';
    }
 
    // A spot-check
    const long double p = 0.123; // a random value in [0, 1]
    const long double q = 1 - p;
    const long double π = std::numbers::pi_v<long double>;
    std::cout << "\n\n" << std::setprecision(19)
              << "β(p,1-p)   = " << std::beta(p, q) << '\n'
              << "π/sin(π*p) = " << π / std::sin(π * p) << '\n';
}

Output:

Pascal's triangle:
 
                  2
                3   3
              4   6   4
            5  10  10   5
          6  15  20  15   6
        7  21  35  35  21   7
      8  28  56  70  56  28   8
    9  36  84 126 126  84  36   9
 
β(p,1-p)   = 8.335989149587307836
π/sin(π*p) = 8.335989149587307834

External links

Weisstein, Eric W. "Beta Function." From MathWorld — A Wolfram Web Resource.

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